When I was not syncing but not getting any errors, I could not access OMA as I would get an error message that indicated it was not working (sorry, I don't remember specifically what it said). That's when I figured out the solution that I documented here:
http://discussions.apple.com/thread.jspa?threadID=1629194&tstart=15
Once I had completed this, everything worked and the iPhone synced and I started getting messages pushed as well. Your ability to log in to OMA successfully makes me think that everything is fine on the Exchange server side, but you may have an issue on the iPhone side. I'd try a couple of things to see what happens. First, it wouldn't hurt to go through the Exchange server and verify the settings that are in my other post, above. Second, I'd delete the iPhone's Exchange account and recreate it to see if that might fix it up.
Finally, if none of these things work, I would try using the domain name with your User ID in the iPhone setting as in:
DomainName\UserID which would look something like this, MyCompany\jane.doe
Also, be sure that you're using a backslash ( \ ) between the domain name and the user ID and not a regular or forward slash ( / ). If you're already using the domain name, try taking it off and only using your user ID.
It would also help to know which version of Windows Server you are running. SBS or Regular.